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\begin{eqnarray*}
\lambda_1^{i,j} &=& \frac{1}{2}(\lambda_1^i + \lambda_1^j)\\
\lambda_2^{i,j} &=& \frac{1}{2}(\lambda_2^i + \lambda_2^j)\\
A_{i,j} &=& (A_{i}A_{j})^{1/2}\\
B_{i,j} &=& \chi_{ij}(B_{i}B_{j})^{1/2}\\
R_{i,j} &=& (R_{i}R_{j})^{1/2}\\
S_{i,j} &=& (S_{i}S_{j})^{1/2}\\
\end{eqnarray*}                           

\end{document}